Trigonometric Leveling: Base not accessible case

TITLE
Determination of RL of the Object Whose Base is Not Accessible

THEORY
Trigonometric leveling is used to determine the reduced level (RL) of an object when its base is inaccessible. This method involves two instrument stations (P and Q) and angular measurements to compute the RL of the object’s top. The key steps involve:

1. Measuring vertical angles (α₁, α₂) from P and Q to the object’s top.
2. Using staff readings (s₁, s₂) at a benchmark (BM) to account for instrument heights.
3. Calculating the horizontal distance (D) from P to the object’s base using the distance (b) between P and Q.
4. Determining the RL of the object’s top using trigonometric principles and averaging the results obtained from both instrument stations.
   

INSTRUMENTS REQUIRED

• Theodolite
• Leveling staff
• Measuring tape
• Tripod stand
• Plumb bob

PROCEDURE

1. Set up the theodolite at point P, level it, and focus the telescope. Set the vertical angle to zero and take the staff reading s₁ at the benchmark (BM).
2. Sight the top of the object from P and record the vertical angle α₁.
3. Rotate the theodolite vertically to align the crosshairs with the top of the object. Mark a point Q on the ground using a plumb bob such that P, Q, and the object’s top are colinear. Measure the horizontal distance b between P and Q.
4. Move the theodolite to Q, set it up, and level it. Set the vertical angle to zero and take the staff reading s₂ at the BM.
5. Sight the top of the object from Q and record the vertical angle α₂.
6. Compute the horizontal distance D from P to the base of the object using the observed values of s₁, s₂, b, α₁, and α₂.
7. Calculate the RL of the object’s top using the first method based on readings from P and the second method based on readings from Q.
8. Compute the average RL of the object’s top by taking the mean of the two calculated RL values.

OBSERVATIONS

CALCULATION

1. Compute D:

   $$D = \frac{(s₂ - s₁) - b \tan \alpha₂}{\tan \alpha₁ - \tan \alpha₂}$$

2. Compute RLₜₒₚ⁽¹⁾:

   $$RL_{\text{top}}^{(1)} = RL_{\text{BM}} + s₁ + D \tan \alpha₁$$

3. Compute RLₜₒₚ⁽²⁾:

   $$RL_{\text{top}}^{(2)} = RL_{\text{BM}} + s₂ + (D + b) \tan \alpha₂$$

4. Compute Average RL:

   $$RL_{\text{top (average)}} = \frac{RL_{\text{top}}^{(1)} + RL_{\text{top}}^{(2)}}{2}$$

RESULTS AND CONCLUSIONS

The average RL of the object’s top is:$\mathrm{<span\; style="color:\; var(--tw-prose-body);\; font-family:\; Inter,\; "Inter\; Fallback";\; font-size:\; 1rem;"></span>}$ _____m. This practical demonstrates the effective use of trigonometric leveling principles to determine the RL of an object whose base is inaccessible. The accuracy of results depends on precise instrument handling, accurate angle measurements, and careful compuations.